From the multivariable generalization of (2), we have, using
(10),
,
where `' is for right-going and `' is for left-going. Thus,
following the classical definition for the scalar case, the wave impedance
is defined by

and we have

(13)

Thus, the wave impedance is the factor of proportionality
between pressure and velocity in a traveling wave. It is diagonal if and
only if the massmatrix is diagonal (since
is assumed
diagonal). The minus sign for the left-going wave
accounts for the
fact that velocities must move to the left to generate pressure to the
left. The wave admittance is defined as
.

More generally, when there is a loss represented by a diagonal matrix
,
we have, in the continuous-time case,

A linear propagation medium in the discrete-time case is completely
determined by its wave impedance (generalized here to
permit frequency-dependent and spatially varying wave impedances). A waveguide is defined for purposes of this paper as a length of medium in
which the wave impedance is either constant with respect to spatial
position , or else it varies smoothly with in such a way that there
is no scattering (as in the conical acoustic tube). For simplicity, we
will suppress the possible spatial dependence and write only
.3